Asymptotic Analysis of Target Fluxes in the Three-Dimensional Narrow Capture Problem

We develop an asymptotic analysis of target fluxes in the three-dimensional (3D) narrow capture problem. The latter concerns a diffusive search process in which the targets are much smaller than the size of the search domain. The small target assumption allows us to use matched asymptotic expansions and Green's functions to solve the diffusion equation in Laplace space. In particular, we derive an asymptotic expansion of the Laplace transformed flux into each target in powers of the non-dimensionalized target size $\epsilon$. One major advantage of working directly with fluxes is that one can generate statistical quantities such as splitting probabilities and conditional first passage time moments without having to solve a separate boundary value problem in each case. However, in order to derive asymptotic expansions of these quantities, it is necessary to eliminate Green's function singularities that arise in the limit $s\rightarrow 0$, where $s$ is the Laplace variable. We achieve this by considering a triple expansion in $\epsilon$, $s$ and $\Lambda\sim \epsilon /s$. This allows us to perform partial summations over infinite power series in $\Lambda$, which leads to multiplicative factors of the form $\Lambda^n/(1+\Lambda)^n $. Since $\Lambda^n/(1+\Lambda)^n \rightarrow 1$ as $s\rightarrow 0$, the singularities in $s$ are eliminated. We then show how corresponding asymptotic expansions of the splitting probabilities and conditional MFPTs can be derived in the small-$s$ limit. Finally, we illustrate the theory by considering a pair of targets in a spherical search domain, for which the Green's functions can be calculated explicitly.